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Asha Crasta et al Int. Journal of Engineering Research and Applications www.ijera.com

ISSN : 2248-9622, Vol. 4, Issue 3( Version 2), March 2014, pp.32-38

www.ijera.com 32 | P a g e

Stability Derivatives of a Delta Wing with Straight Leading Edge

in the Newtonian Limit

Asha Crasta*, S. A. Khan**

*(Research Scholar, Department of Mathematics, Jain University, Bangalore, Karnataka, India)

**(Principal, Mechanical Engineering Department, Bearys Institute of Technology, Innoli Mangalore,

Karnataka, India)

ABSTRACT This paper presents an analytical method to predict the aerodynamic stability derivatives of oscillating delta

wings with straight leading edge. It uses the Ghosh similitude and the strip theory to obtain the expressions for

stability derivatives in pitch and roll in the Newtonian limit. The present theory gives a quick and approximate

method to estimate the stability derivatives which is very essential at the design stage. They are applicable for

wings of arbitrary plan form shape at high angles of attack provided the shock wave is attached to the leading

edge of the wing. The expressions derived for stability derivatives become exact in the Newtonian limit. The

stiffness derivative and damping derivative in pitch and roll are dependent on the geometric parameter of the

wing. It is found that stiffness derivative linearly varies with the pivot position. In the case of damping

derivative since expressions for these derivatives are non-linear and the same is reflected in the results. Roll

damping derivative also varies linearly with respect to the angle of attack. When the variation of roll damping

derivative was considered, it is found it also, varies linearly with angle of attack for given sweep angle, but with

increase in sweep angle there is continuous decrease in the magnitude of the roll damping derivative however,

the values differ for different values in sweep angle and the same is reflected in the result when it was studied

with respect to sweep angle. From the results it is found that one can arrive at the optimum value of the angle of

attack sweep angle which will give the best performance.

Keywords: Straight leading edges, Newtonian Limit, Strip theory

I. INTRODUCTION Unsteady supersonic/hypersonic

aerodynamics has been studied extensively for

moderate supersonic Mach number and hypersonic

Mach number for small angles of attack only and

hence there is evidently a need for a unified

supersonic/hypersonic flow theory that is applicable

for large as well as small angles of attack.

For two-dimensional flow, exact solutions

were given by Carrier [1] and Hui [2] for the case of

an oscillating wedge and by Hui [3] for an oscillating

flat plate. They are valid uniformly for all supersonic

Mach numbers and for arbitrary angles of attack or

wedge angles, provided that the shock waves are

attached to the leading edge of the body.

For an oscillating triangular wing in

supersonic/hypersonic flow, the shock wave may be

attached or detached from the leading edges,

depending on the combination of flight Mach

number, the angle of attack, the ratio of specific heats

of the gas, and the swept-back angle of the wing. The

attached shock case was studied by Liu and Hui [4]

where as the detached shock case in hypersonic flow

was studied by Hui and Hemdan both are valid for

moderate angles of attack. Hui et.al [5] applied the

strip theory to study the problem of stability of an

oscillating flat plate wing of arbitrary plan form

shape placed at a certain mean angle of attack in a

supersonic/hypersonic stream. For a given wing plan

form at a given angle of attack, the accuracy of the

strip theory in approximating the actual three-

dimensional flow around the wing is expected to

increase with increasing flight Mach Number. The

strip theory becomes exact in the Newtonian limit

since the Newtonian flow, in which fluid particles do

not interact with each other is truly two-dimensional

locally. In this paper the Ghosh theory [6] is been

used and the relations have been obtained for the

stability derivatives in pitch and roll in the

Newtonian limit.

II. Analysis Consider a wing with a straight leading

edge.

The Stiffness Derivative is given by

1

2sin cos ( )( )0 0

3C f S hm

(1)

Damping derivative in pitch is given by

0 1

4 12sin ( )( )3 2q

C f S h hm (2)

RESEARCH ARTICLE OPEN ACCESS




Where

1

1 2 2 2( ) [2 ( 2 ) / ( ) ]1 1 1 12 1

f S s B S B SS

sin1S M

4 2( )

1B

in all above cases.

In the Newtonian limit M tends to infinity and

tends to unity. In the above expression (1) and (2)

only )( 1Sf contains M and .

1

1

2

11 1 1

21 21

2

1

2 21 1

( 2 )( 1)lim ( ) lim {2 }

2( )

(4 2 ) lim 2 4

(4 )

M M

S

B Sf S S

SB S

S

S s

(3)

Therefore in the Newtonian limit, stiffness

derivative in pitch,

0

22( )

sin 2 3

mCh

(4)

The Damping derivative in Newtonian limit

for a full sine wave is given by

0

2 4 14sin ( )

3 2qmC h h (5)

We define g(h) = 2 4 1

( )3 2

h h which is a

quadratic in pivot position h and hence has a

minimum value

4sin ( )qmC g h

In Eq. (5), only g (h) depends on h and other terms

are constant. To get minimum value of qmC only

g (h) is to be differentiated and putting )(hgh

equal to zero

2 4 1[( )] 0

3 2h h

h

2

3h

Let the value for h corresponding to

min][qmC be

denoted mh .

mh =2

3

Hence min)(hg 2 4 1( )

3 2m mh h

qmC = min4sin ( )g h

sin

minqmC

= min4 ( )g h

(6)

Rolling Moment due to roll in Newtonian

limit becomes

04sin cot

12plC

0

cot

sin 3

plC

(7)

RESULTS AND DISCUSSION The main purpose of this study is to find

stiffness and damping derivatives in pitch of a delta

wing with straight leading edge in the Newtonian

limit where the Mach number will tend to infinity and

the specific heat ratio gamma will tend to one, in this

situation when the stability derivatives are considered

how they vary with the change in the geometric

parameters.

Fig. 1 shows the variation of stiffness

derivatives with the non-dimensional pivot position.

From the figure it is seen that at the leading edge it

has the maximum value of around 1.35 and the

downstream of the leading edge it decreases with

increase in the pivot position where as the minimum

value at the end of the trailing edge is around -0.75. It

is also seen that the location of center of pressure has

also, shifted towards the trailing edge of the wing

otherwise this non-dimensional pivot location is in

the range from 50 to 60 percent from the leading

edge, when they are evaluated for supersonic and

hypersonic Mach numbers, this shift towards the

trailing edge will be advantageous from the static

stability point of view as with this shift of center of

pressure will result in higher values of the static

margin and hence large value of the stiffness

derivative. When the damping derivatives are

considered it is found that there is nonlinearity in the

damping derivative with respect to the pivot position

in Newtonian limit as shown in Fig. 2. From the

figure it is seen that at the leading the magnitude of

the damping derivative is 2 and at the trailing edge is

around 0.75. If we compare these results of the

damping derivatives with that for low supersonic and

hypersonic Mach numbers, it is found that the initial




value itself is increased by 30 percent that it

continues to be high value for all the values of the

non-dimensional pivot position. Also, it is seen that

the minima of the pitch damping derivative too has

shifted towards the trailing edge resulting in the

higher values of the pitch damping derivative and

hence will perform better in the dynamic conditions

as compared to those at supersonic and hypersonic

Mach number. The physical reasons for this trend

may be due to the change in the pressure distribution

on the wing plan form. Results of stiffness derivative

as a function of angle of attack are shown in Fig. 3,

for a fixed pivot position of h = 0 and h = 0.6.

From the figure it is seen that the Stiffness increases

linearly with angle of incidence. It is also seen that

stiffness derivative assumes very high value when it

is computed at the leading edge and the center of

pressure and these assumes importance while

analyzing the performance of the wing, as stiffness

derivative is a measure of static stability. At the

design stage depending upon the requirement we

need to work the balance between the upper and

lower limit of the static margin.

Fig. 4 shows the variation of the damping

derivatives with respect to angle of attack for three

pivot positions namely h = 0, 0.6, and 1.0, this was

considered just ascertain the capability of the wing

under three different situations. It is well known that

stiffness derivative ensures the static stability of the

wing, but if the system is statically stable that doesn’t

mean that it is dynamically stable as well; hence

damping derivative assumes a very important role in

the analysis of the stability and control. From the

figure it is also, seen that damping derivative varies

linearly for all the values of the pivot position. The

reasons for this behavior may due the computational

domain as they are computed at the locations of the

wing leading edge, the center of pressure, and at the

trailing edge; where the location has been fixed and

the effect of the angle of incidence alone is being

reflected in these results.

Results for roll damping derivative as a

function of angle of incidence are shown in Figs. 5 to

7. From the figure it is seen that roll damping

derivatives, also varies linearly with respect to the

angle of attack. When the variation of roll damping

derivative was considered, it is found that it also,

varies linearly with angle of attack for given sweep

angle, but with increase in sweep angle there is

continuous decrease in the magnitude of the roll

damping derivative however, the values differ for

different values in sweep angle and the same is

reflected in the result when it was studied with

respect to sweep angle.

Fig. 5 shows results for roll damping

derivative variation with angle of incidence for sweep

angle in the range five degrees to thirty degrees. It is

found that the difference in the values of the roll

damping derivative is very high between sweep

angles from five to 10 degrees, then this decrease in

the magnitude is gradual. The reason for this

behavior may be due to the presence of the ratio of

cosine and sine term; and for zero sweep angle the

value of cosine term is one, whereas sine term is

zero, and this value is getting reversed when sweep

angle is ninety degrees. The similar trends are

reflected in Figs. 6 and 7 when sweep angle in the

range from thirty five to eighty degrees.

Figs. 8 and 9 present the results of roll

damping derivative as a function of angle of attack. It

is seen that when angle of attack is in the range from

five degrees to fifteen degrees optimum sweep angle

may be in the range ten degrees to twenty degrees as

seen in figure 8, where if the angle of attack range is

between twenty to thirty degrees then sweep angle in

the range from fifteen degrees to fifty degrees may

the best option but one has to analyze the further,

keeping in mind that whether; for the given

combination, the leading edge of the wing will be

sub-sonic or supersonic. Hence, one has to consider

all these parameters to arrive at the optimum solution.

III. Conclusion The present theory is valid when the shock

wave is attached to the leading edge. The effect of

secondary wave reflections and viscous effects are

neglected. The expressions derived for stability

derivatives become exact in the Newtonian limit.

From the results it is found that the stability

derivatives are independent of Mach number as they

are estimated in the Newtonian limit, where Mach

numbers will tend to infinity and specific heat ratio

gamma will tend to unity. The stiffness derivative

and damping derivative in pitch they reflect only the

effect of the geometric parameter of the wing in the

Newtonian limit too. It is found that stiffness

derivative linearly varies with the pivot position as

the same was found for the cases in our previous

results at low supersonic, supersonic, and Hypersonic

Mach numbers, however, the location of center of

pressure has shifted towards the trailing edge by

nearly twenty percent which; will result in increased

value of the static margin and hence increase value of

stiffness derivative resulting in better performance in

the dynamic conditions. In the case of damping

derivative since the expression for the damping

derivative is non-linear and the same has been

reflected in the results. For damping derivative also,

the initial value also has increased followed by

shifting of center of pressure towards the trailing

edge resulting in better performance of the plane.

Roll damping derivatives also vary linearly with

angle of attack for a given value of sweep angle;

however, for a given angle attack when roll damping




derivatives were plotted with sweep angle the

behavior is non-linear. From the above results it can

be stated that if angle of attack range is between ten

degrees to fifteen degrees then the optimum sweep

angle will be in the range five to twenty degrees,

however, if the angle of attack is in the range of

twenty to thirty degrees then the optimum range of

the sweep angle will be between fifteen to fifty

degrees.

Fig. 1: variation of stiffness derivative in pitch with pivot position

Fig. 2: variation of damping derivative in pitch with pivot position




Fig. 3: Variation of damping derivative in pitch with angle of incidence

Fig. 4: variation of stiffness derivative in pitch with angle of incidence

Fig. 5: variation of Roll damping derivative with angle of incidence




Fig. 6: variation of roll damping derivative with angle of incidence

Fig. 7: variation of roll damping derivative with angle of incidence

Fig. 8: variation of roll damping derivative with Sweep angle




Fig. 9: variation of roll damping derivative with Sweep angle

References [1] Carrier G. F., The Oscillating Wedge in a

Supersonic Stream, Journal of the

Aeronautical Sciences, Vol.16, March

1949, pp.150-152.

[2] Hui W. H., Stability of Oscillating Wedges

and Caret Wings in Hypersonic and

Supersonic Flows,AIAA Journal, Vol.7,

August 1969, pp. 1524-1530.

[3] Hui. W. H, Supersonic/Hypersonic Flow

past an Oscillating Flat plate at Large

angles of attack, Journal of Applied

Mathematics and Physics, Vol. 29, 1978,

pp. 414-427.

[4] Liu D. D and Hui W. H., Oscillating Delta

Wings with attached Shock waves,AIAA

Journal, Vol. 15, June 1977, pp. 804-812.

[5] Hui W. H. et al, Oscillating Supersonic

Hypersonic wings at High Incidence,

AIAA Journal, Vol. 20, No.3, March1982,

pp. 299- 304.

[6] Asha Crasta and Khan S. A, Oscillating

Supersonic delta wing with Straight

Leading Edges, International Journal of

Computational Engineering Research, Vol.

2, Issue 5, September 2012, pp.1226

1233,ISSN:2250-3005.

[7] Asha Crasta and Khan S. A., High

Incidence Supersonic similitude for Planar

wedge, International Journal of

Engineering research and Applications,

Vol. 2, Issue 5, September-October 2012,

pp. 468-471, ISSN: 2248-9622

[8] Asha Crasta, M. Baig, S.A. Khan,

Estimation of Stability derivatives of a

Delta wing in Hypersonic flow,

International Journal of Emerging trends in

Engineering and Developments in Vol.6,

Issue2, Sep 2012, pp505-516, ISSN:2249

6149.

[9] Asha Crasta, S. A. Khan, Estimation of

stability derivatives of an Oscillating

Hypersonic delta wings with curved

leading edges, International Journal of

Mechanical Engineering & Technology,

vol. 3, Issue 3, Dec 2012, pp. 483-492.

[10] Asha Crasta and S. A. Khan, Stability

Derivatives in the Newtonian Limit, The

International Journal of Advanced

Research in Engineering and Technology,

Volume: 4, Issue: 7, Nov-Dec 2013, pp

.276-289.

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